Azarenka of Belarus also saved four match points but committed 12 double faults to none for Pegula, a top seed for only the sixth time in her career.
Pegula advanced to the semi-finals against fourth-seeded Daria Kasatkina, a Russian who converted 10 of 20 break points in topping Romania's Jaqueline Cristian 6-7(4), 6-2, 6-3 in a match lasting two hours and 42 minutes.
Danielle Collins won her 11th consecutive match to make it two Americans in the Charleston semi-finals for the first time since sisters Venus and Serena Williams were in the final four 11 years ago.
The unseeded Collins knocked off No. 11 Elise Mertens of Belgium 6-3, 6-4 in one hour and 24 minutes to advance to her 15th career tour-level semi-final.
Collins, a 30-year-old Floridian who is retiring at the end of the season, captured the Miami Open championship last week.
The other quarter-final match pitted third-seeded Maria Sakkari of Greece against ninth-seeded Veronika Kudermetova of Russia.
Copa Colsanitas
Eighth-seeded Kamilla Rakhimova of Russia was shut out in the first set, then rallied past fourth-seeded Cristina Bucsa of Spain 0-6, 6-4, 7-5 in the quarter-finals at Bogota, Columbia.
Sara Errani of Italy also got off to a slow start before eliminating Irina Bara of Romania 4-6, 6-2, 6-3 and advancing to the semi-finals. Errani saved 17 of 23 break points while converting 8 of 20 opportunities.
In other quarter-final matches Friday night, top-seeded Marie Bouzkova of the Czech Republic faced seventh-seeded Laura Siegemund of Germany; and home-country favorite Camila Osorio, the sixth seed, played second-seeded Tatjana Maria of Germany.